Question: Find an explicit formula for the geometric sequence $12\,,\,24\,,\,48\,,\,96,...$. Note: the first term should be $\textit{b(1)}$. $b(n)=$
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{96}{48}=\dfrac{48}{24}=\dfrac{24}{12}={2}$ We see that the constant ratio between successive terms is ${2}$. In other words, we can find any term by starting with the first term and multiplying by ${2}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $f(n)$ ${12}\cdot\!{2}^{\,0}$ ${12}\cdot\!{2}^{\,1}$ ${12}\cdot\!{2}^{\,2}$ ${12}\cdot\!{2}^{\,3}$ We can see that every term is the product of the first term, ${12}$, and a power of the constant ratio, ${2}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${12}$ is the first term and ${2}$ is the constant ratio): $b(n)={12}\cdot{2}^{{\,n-1}}$ Note that this solution strategy results in this formula; however, an equally correct solution can be written in other equivalent forms as well.